Oh they most certainly are. There was a reason druids could turn into mundane earth animals but not displacer beasts and owlbears (prior to the movie at least). The game used to keep some notion that things that also exist in the real world act like they do in the real world, and if they don't exist in the real world there is a magical explanation for it. Gods, demon lords, "a wizard did it". It gives the world a frame of reference that without makes the world alien.
But if that's what people want, go nuts. Cats speak common. Horses run on air. Wolves breathe underwater, humans can jump 30 feet vertical leap. Because: dragons.
I feel you are being a bit dramatic and overlooking some powers that already exist in the game.
Let's look at
- evasion: a rogue can take 0 damage when in the epicentre of a fireball or meteor swarm. They can do this routinely, without the benefit of cover and even when there's no possible place to escape the explosion(like trapped a small room). Sounds like magic.
-blindsense: a rogue is aware of invisible enemies within 10 feet as long as they can hear. They must have some kind of supernatural hearing.
-Stroke of Luck: can defy odds and turn a miss into a hit. Without needing to be the progeny of the luck god.
-Second wind: can (non)magically heal themselves without needing a cleric
-Extra attack+actions surge: attack 1/second without needing to be magically hasted. Without being part speedy wind elemental.
Those are all base class abilities.
Horses run on air: they are called pegasus. (also non-magical creatures). Nightmares do it too.
Cats speak common: Tabaxi
I haven't found a Wolf that breathes water but a Winter Wolf can breath a cone of cold.
I won't argue this point anymore. Obviously, in your fiction or fantasy settings, you need to justify specific abilities with some kind of mythical background. It just sounds kind of random to me since so many abilities are clearly unrealistic and don't need the same kinds of justifications.
For me, at least, I'd be happy - to go back to my original point - to have martial classes get a few epic abilities based on their mundane skills to represent how they've perfected those skills to such an extent that they become supernatural. I'd be happy to have that so that they could compete with wizardy classes and I'd be happy to let them have it without needing to codify the fluff as part of the class.